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\title{\huge Archimedean Spirals \footnote{This file is from the 3D-XploreMath project.}}
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\centerline{\includegraphics{archimedean_spirals.png}}


Archimedean spiral is defined by the polar equation $r = \theta^n$.
Special names are given for some value of n.


\begin{enumerate}
\item
if $n = 1$, we have $r = \theta$, it's called Archimedes' spiral.
% This spiral is easily recognized because it cuts a radial line from the pole into equal segments. need to make sure this statement is true. need to prove it, esp near the center.
\item
if $n = -1$, we have $r = 1/\theta$, it's called reciprocal spiral. (aka hyperbolic spiral)
\item
if $n= 1/2$, we have $r = \sqrt{\theta}$, it's called Fermat's spiral. (aka parabolic spiral).
\item
if $n= -1/2$, we have $r = 1/\sqrt{\theta}$, it's called Lituus. 
\end{enumerate}


The reason parabolic spiral and hyperbolic spiral are so named is because their equation in polar system $r \theta=1$ and $r^2=\theta$ resembles the equation for hyperbola $x y=1$ and parabola $x^2=y$ in rectangular coordinates system.

% to do: (*Question: What's Fermat's involvement with parabolic spiral?*) 

Hyperbolic spiral is also called reciprocal spiral because it is the inverse curve of Archemedes' spiral, with inversion center at the center.

% to do: I forgot where did i get this. It may also refer to the fraction $1/\theta$.

% to do: The word Lituus is derived from greek or latin, meaning crooked, spiral?.

The inversion curve of all Archemedean spirals with respect to a circle on center is another Archemedean spiral. This is easily proven as follows.

To convert a polar formula $r=f(\theta)$ to parametric formula, all we have to do is to multiply $f$ to $(\cos \theta, \sin \theta)$. So, the Archimedean spiral in parametric form is $(t^n \cos t, t^n \sin t)$. The inversion at origin with radius b of a point $(x,y)$ is
 $\{b^2 x, b^2 y\}/ (x^2 + y^2)$. Apply this to the parametric form and simply we get $b^2 (\cos(t) t^{-n}, \sin(t) t^{-n})$, which in polar form is $r=b^2 \theta^{-n}$. When b==1, there's no scaling. 

From the above, we can see that the Archimedes' spiral inverts to the reciprocal spiral, and Fermat's spiral inverts to the Littus.

The following two images illustrates Archimedes's spiral and Reciprocal spiral as mutual inverses. The red curve is the reciprocal spiral, the purple is the Archimedes' spiral. The yellow is the inversion circle. 
\centerline{\includegraphics{inv1.png}}
\centerline{\includegraphics{inv2.png}}

The following image illustrates a Lituus and Fermat's spiral as mutual inverses. The red curve is the Fermat's spiral. The blue curve is its inversion, which is a lituus scaled by $5^2$. The yellow circle is the inversion circle with radius 5. Note that points inside the circle gets mapped to outside of the circle. The closer the point is to the origin, the farther is its corresponding point outside the circle.

\centerline{\includegraphics{lituus_inv.png}}

% dick, is this graphics too big in pdf? probably i need to generate a smaller version, but i wanted to see your opinion. .. as i'm not clear how bitmapped images are embedded into vector-based tex/PDF... perhaps a rescaling command in TeX would work ok?

% PS several images in this pdf are probably too big. The first image is a composition of 4. I made it into one since i'm not sure how to arrange 4 pngs into such an array...

% anyway, please let me know what do you think.


XL.


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